Regularity Assumptions:  In this work, we assume our outer functions are \(\mu _{\ell }\) -weakly convex, bounded, Lipschitz, and Lipschitz-continuous, while we assume our inner functions are Lipschitz continuous and \(\mu _g\) -strongly convex. We use the same assumption here as in [1]}, with the addition of:
\(\mathbb {E} [h_{\lambda }^k] = \nabla _{\lambda } F(x_k, \lambda _k) = \left[ f_1(x_k, y_{k+1}^1), \cdots , f_n(x_k, y_{k+1}^n) \right]^\top . \) 
